TITLE:
Theoretical Model of Diffraction Line Profiles as Combinations of Gaussian and Cauchy Distributions
AUTHORS:
Girija Bhushan Mitra
KEYWORDS:
Line Broadening, Diffraction Profile, Theoretical Model, Gaussian Function, Cauchy Function, Voigt Function, Pseudovoigt Function, Fourier Transform, Second Moment, Fourth Moment, Fourth Cumulant
JOURNAL NAME:
Journal of Crystallization Process and Technology,
Vol.4 No.3,
July
14,
2014
ABSTRACT:
Previously
we derived equations determining line broadening in ax-ray diffraction profile
due to stacking faults. Here, we will consider line broadening due to particle
size and strain which are the other factors affecting line broadening in a
diffraction profile. When line broadening in a diffraction profile is due to
particle size and strain, the theoretical model of the sample under study is
either a Gaussian or a Cauchy function or a combination of these functions,
e.g. Voigt and Pseudovoigt functions.
Although the overall nature of these functions can be determined by Mitra’s R(x) test and the Pearson and Hartley xtest, details of a predicted model will be lacking. Development of a
mathematical model to predict various parameters before embarking upon the
actual experiment would enable correction of significant sources of error prior
to calculations. Therefore, in this study, predictors of integral width, Fourier
Transform, Second and Fourth Moment and Fourth Cumulant of samples represented
by Gauss, Cauchy, Voigt and Pseudovoigt functions have been worked out. An
additional parameter, the coefficient of excess, which is the ratio of the
Fourth Moment to three times the square of the Second Moment, has been
proposed. For a Gaussian profile the coefficient of excess is one, whereas for
Cauchy distributions, it is a function of the lattice variable. This parameter
can also be used for determining the type of distribution present in aggregates
of distorted crystallites. Programs used to define the crystal structure of
materials need to take this parameter into consideration.